Wave breaking in solutions of the dispersionless Kadomtsev–Petviashvili equation at a finite time

We discuss some interesting aspects of the wave breaking in localized solutions of the dispersionless Kadomtsev–Petviashvili equation, an integrable partial differential equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in $2+1$ dimensions, which arise in several physical contexts such as acoustics, plasma physics, and hydrodynamics. For this, we use an inverse spectral transform for multidimensional vector fields that we recently developed and, in particular, the associated inverse problem, a nonlinear Riemann–Hilbert problem on the real axis. In particular, we discuss how the derivative of the solution blows up at the first breaking point in any direction of the plane $(x,y)$ except in the transverse breaking direction and how the solution becomes three-valued in a compact region of the plane $(x,y)$ after the wave breaking.